Sparse corruption in low-rank matrix inference: the PCA benchmark

Principal Component Analysis (PCA) is a standard tool for extracting a low-rank signal from noisy observations. It is known that applying PCA to a rank-one signal corrupted by a dense, homogeneous noise, in the large matrix size limit, the celebrated BBP transition occurs, where the emergence of an outlying eigenvalue and the alignment of the corresponding eigenvector occur at the same critical signal strength. Here we study the case of sparse noise corruption. The noise matrix is modelled as the adjacency matrix of a weighted undirected graph with finite average connectivity. Using the replica method, we analytically compute the typical top eigenvalue, the top eigenvector component density, and the squared overlap with the signal, through recursive distributional equations solved by population dynamics. We identify two signal-strength transitions as functions of graph connectivity: θ_{\rm crit}, marking signal recovery by the top eigenvector and generalising the BBP transition, and θ_{\rm b}, where the signal-related eigenvalue detaches from the bulk. For noise with nonzero mean, these transitions need not coincide because of a structural sparse-graph outlier, leading to a discontinuous transition in the squared overlap with the top eigenvector. The same top-eigenpair formalism also predicts the overlap of the signal with the eigenvector associated with the second largest eigenvalue when the signal eigenvalue is an outlier but remains below the structural outlier, where the transition is continuous. We specialise the equations to Poissonian and Random Regular degree distributions, recover dense-noise results in the large-connectivity limit, and validate the theory by numerical diagonalisation of large matrices.